Our work mostly aligns with the efforts on obtaining inner approximations of the PSD cone, but proposes a more flexible family of inner approximations based on Grassmannian packing problems. More specifically, we develop a novel decomposition framework of the PSD cone that generalizes the above-mentioned approximation techniques. Using this decomposition, we develop novel inner approximations of the PSD cone, based on conic combinations of smaller dimensional sub-cones, such that they are maximally separated from each other. In doing so, these approximations tend to occupy large fractions of the volume of the PSD cone and provide more accurate inner approximations.
Such problem of placing maximally distant sub-cones is closely related to the classical packing problem in Grassmannian manifolds. The Grassmannian packing problem [1]} aims to answer the following question: how should \(N\)  \(k\) -dimensional subspaces of an \(n\) -dimensional Euclidean space be arranged so that they are as far apart as possible? The problem of finding Grassmannian packings is a fundamental problem of geometry and has applications in many fields such as signal processing [2]} , information theory [3]}, [4]}, and wireless communications [5]}. The hope is that by establishing these connections, this work will ignite a new search for better approximations that exploit classical tools from, e.g., information theory.
